The M\"obius function

The Möbius function $\mu (n)$ is defined as $(-1)^r$ if $n$ is a product of $r$ distinct primes, and as $0$ if the square of a prime divides $n$. Define :

$$M(x) := \sum_{n \leq x} \mu (n).$$

Then RH is equivalent to each of the following statements

$$M (x) \ll x^{1/2 + \epsilon},$$

for every positive $\epsilon$ ;

$$M (x) \ll x^{1/2} \exp (A \log x/\log\log x);$$

for some positive A.

Both results are due to Littlewood.

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